Global existence for fully nonlinearreaction-diffusion systems describing
multicomponent reactive flow
Martine MARION, Ecole Centrale de Lyon
Dynamics and Differential Equations, IMA, June 2016
Dedicated to the memory of George Sell
1 - The governing equations
2 - The diffusive fluxes : mathematical study
3 - The fundamental energy estimate
4 - Existence of weak solutions
Joint work with R. Temam
1 - The governing equations
• Unknowns
v ,p : velocity and pressure
θ : (reduced) temperature
Y = (Y1, . . . ,YN) : mass fractions of the N species.
• Equations for hydrodynamics and combustion
∂v∂t
+ (v ·∇)v − Pr ∆v + ∇p = edσθ
div v = 0
∂θ
∂t+ (v ·∇)θ −∆θ = −
N∑i=1
hiωi(θ,Y1, . . . ,YN)
∂Yi
∂t+ (v ·∇)Yi + ∇ · Fi = ωi(θ,Y1, . . . ,YN), 1 ≤ i ≤ N
1 - The governing equations
• Chemical rates : Arrhenius law
R reversible reactions :∑N
i=1 νji Si ↔
∑Ni=1 λ
jiSi , 1 ≤ j ≤ R
Production rate of species i : ωi =∑R
j=1(λji − ν
ji )rj
Reaction rate of reaction j : rj = K dj∏N
i=1( YiMi
)νji − K r
j∏N
i=1( YiMi
)λji
• Mathematical properties
N∑i=1
ωi(θ,Y ) = 0 for θ ≥ 0, 0 ≤ Yk ≤ 1
ωi(θ,Y ) = αi(θ,Y )− Yiβi(θ,Y ), αi ≥ 0, βi ≥ 0, boundedN∑
i=1
hiωi(0,Y ) ≤ 0
1 - The governing equations
Diffusion in gaseous mixtures
∂Yi∂t + (v ·∇)Yi + ∇ · Fi = ωi(θ,Y1, . . . ,YN), 1 ≤ i ≤ N
Fi = YiVi Vi = diffusion velocity of species i = v i − v
•∑N
i=1 YiVi = 0
•∇Xi =∑N
j=1,j 6=iκDij
XiXj(Vj − Vi) i = 1, . . . ,N
Stefan-Maxwell equations
Xi = mole fraction
Dij = Dji > 0 binary diffusion coefficient for species i and j
If Dij = D for all i, j :
∇Xi = − κD Fi i = 1, . . . ,N Fick’s Law
1 - The governing equations
• Mathematical studies
Local in time solutions or simplified models :
Giovangigli-Massot (1998)Bothe (2011), Boudin-Grec-Salvarani (2012), Juengel-Stelzer (2013),Herberg-Meyries-Prüss-Wilke (2014)
Chen-Juengel (2015)
2 - The diffusive fluxes : mathematical study
• Diffusion equations
∂Yi∂t + (v ·∇)Yi + ∇ · Fi = ωi(θ,Y1, . . . ,YN), 1 ≤ i ≤ N
Fi = YiVi where
∑Ni=1 YiVi = 0∑Nj=1,j 6=i
κDij
XiXj(Vj − Vi) = ∇Xi i = 1, . . . ,N
• Determination of the diffusion velocities when Yi > 0, ∀i
Y − X relations : Xi = YiMi
1YM
where YM =∑N
j=1YjMj
(N + 1) equations :
∑Ni=1 YiVi = 0
B(Y )V = −Y 2M∇X
Bij(Y ) =
−d′ijYiYj for j 6= i,
N∑k=1;k 6=i
d′ikYiYk for j = i,d′ij = κ
Dij Mi Mj
2 - The diffusive fluxes : mathematical study ∑Ni=1 YiVi = 0
B(Y )V = −Y 2M∇X
• The matrix B(Y ) is symmetric semi-definite positive
• For an appropriate value of γ > 0 the matrix
Cij(Y ) = Bij(Y ) + γYiYj , 1 ≤ i, j ≤ N
is symmetric positive definite∑Ni,j=1 Cij(Y )Vj · Vi ≥ γ
(∑Nj=1 Yj
)(∑Ni=1 Yi |Vi |2
)• The N + 1 equations are equivalent to
C(Y )V = −Y 2M∇X
2 - The diffusive fluxes : mathematical study
• Determination of the fluxes when Yi ≥ 0 ∀i,∑N
i=1 Yi > 0Fi = YiVi if Yi > 0Fi = 0 if Yi = 0
Fi = −∑N
j=1 aij(Y1, . . . ,YN)∇Yj , aij = rational functions of Y1, . . . ,YN
• Diffusion equations
∂Yi∂t + (v ·∇)Yi + ∇ · Fi = ωi(θ,Y1, . . . ,YN), 1 ≤ i ≤ N
Boundary conditions :
x ∈ Ω = (0, `)× (0,h) or (0, `)× (0, L)× (0,h)
Yi = Y ui > 0 at xd = 0, ν · Fi = 0 otherwise
Initial conditions : Yi(x , 0) = Yi,0(x) ≥ 0
3 - The fundamental energy estimate
∂Yi∂t + (v ·∇)Yi + ∇ · Fi = ωi(θ,Y1, . . . ,YN), 1 ≤ i ≤ N
We suppose Yi > 0 ∀i and∑N
i=1 Yi = 1. Then µi = 1Mi
log Xi is defined.
µi − µui = ∂g
∂Yi, g(Y ) =
∑Nj=1
YjMj
[log Xj − log Xu
j
]entropy of the mixture
g is convex
• ddt
∫Ω
g(Y ) +∫
Γhg(Y )−
∑Ni=1
∫Ω
F i ·∇µi =∑N
i=1
∫Ωωi(θ,Y )(µi − µu
i )
• Thanks to ωi(θ,Y ) = αi(θ,Y )− Yiβi(θ,Y )
ωi(θ,Y )(µi − µui ) ≤ c
• −∑N
i=1
∫Ω
F i ·∇µi ≥ c1∫
Ω|∇Y |2
• ddt
∫Ω
g(Y ) + c1∫
Ω|∇Y |2 ≤ c
3 - The fundamental energy estimate
• −∑N
i=1
∫Ω
F i ·∇µi ≥ c1∫
Ω|∇Y |2
Fi = YiVi , C(Y )V = −Y 2M∇X , Yi
Mi Xi= YM
−∑N
i=1 Fi ·∇µi = −∑N
i=1 YiVi · ∇XiMi Xi
= 1YM
∑Ni,j=1 Cij(Y )Vj · Vi
−∑N
i=1 Fi ·∇µi ≥ γ(∑N
i=1 Yi |Vi |2)
≥ c|∇X |2
≥ c1|∇Y |2
4 - Existence of solutions
Introduction of normal unknowns
Kawashima-Shizuta (1988), Giovangigli-Massot (1996),Degond-Genieys-Juengel (1997)
Ei : ∂Yi∂t + (v ·∇)Yi + ∇ · Fi = ωi(θ,Y ) ; E1 + ....+ EN = 0
∑Ni=1 Ei
log XiMi
=∑N−1
i=1 Ei ( log XiMi−
log(1−∑N−1
j=1 Xj )
MN)
ξi = log XiMi−
log(1−∑N−1
j=1 Xj )
MN, i = 1, ...,N − 1 (∗)
• For ξ ∈ RN−1 the relations (∗) determine uniquely X = (X1, . . . ,XN)
with 0 < Xi < 1 ∀i and∑N
i=1 Xi = 1
• Y ∈ RN can be expressed as C∞ function of ξ ∈ RN−1 with 0 < Yi < 1
and∑N
i=1 Yi = 1 ; we set : Y = Y(ξ)
4 - Existence of solutions
• Equations for the unknown ξ = (ξ1, ξ2, ..., ξN−1) :
∂Yi∂t + (v ·∇)Yi + ∇ · Fi = ωi(θ,Y ), 1 ≤ i ≤ N − 1
• Time discretization with time step k > 0 :
1k (Y m
i − Y m−1i ) + (vm ·∇)Y m
i +∇ · Fmi −k∆ξm
i = ωi(θm,Y m), 1 ≤ i ≤ N − 1
The ξm are defined recursively and Y m = Y(ξm). Also
vm(x) = 1k
∫ mk(m−1)k v(x , t)dt , θm(x) = 1
k
∫ mk(m−1)k θ(x , t)dt
• Boundary conditions :
ξmi = ξu
i at xd = 0
4 - Existence of solutions
1k (Y m
i −Y m−1i ) + (vm ·∇)Y m
i +∇ · Fmi − k∆ξm
i = ωi(θm,Y m), 1 ≤ i ≤ N− 1
ξmi = ξu
i at xd = 0
• Existence of solution by the Galerkin method
A priori estimates = discrete analogs of the fundamental energyestimate∫
Ωg(Y m)−
∫Ω
g(Y m−1) + c1k∫
Ω|∇Y m|2 + k2 ∫
Ω|∇ξm|2 ≤ ck
• Passage to the limit k → 0 : add equation for Y mN = 1−
∑N−1i=1 Y m
i
1k (Y m
i − Y m−1i ) + (vm ·∇)Y m
i + ∇ · Fmi −k∆ξm
i = ωi(θm,Y m), 1 ≤ i ≤ N
ξNi = −
∑N−1i=1 ξi
4 - The full system
Add equations for the velocity, pressure and temperature :
1k (Y m
i −Y m−1i ) + (vm ·∇)Y m
i +∇ · Fmi − k∆ξm
i = ωi(θm,Y m), 1 ≤ i ≤ N− 1
1k (vm − vm−1) + (vm ·∇)vm − Pr ∆vm + ∇pm = edσθ
m, divvm = 0,
1k (θm − θm−1) + (vm ·∇)θm −∆θm = −
∑Ni=1 hiωi(θ
m+,Y m)
Existence of weak solutions for the equations coupled with thehydrodynamics and temperature equations
